Method for scheduling the operation of power generators using factored Markov decision process

ABSTRACT

An optimal conditional operational schedule for a set of power generators is determined by constructing states and transitions of a factored Markov decision process (fMDP) from a target electrical demand and generator variables. A cost function for the fMDP is constructed based on the electrical demand, the generator variables, and a risk coefficient. Then, the fMDP is solved to obtain the optimal conditional operational schedule.

FIELD OF THE INVENTION

The field of the invention relates generally to power generation, andmore particularly to scheduling the operation of power generators.

BACKGROUND OF THE INVENTION

It is desired to schedule the operation of power generators, e.g.,nuclear, coal, oil, gas, hydroelectric, solar, and wind. The generatorsare connected to consumers via electrical grids. The grids can covercontinents. An objective of the operational scheduling is to generate aprojected amount of electrical power for the consumers, while minimizingproduction cost and risk of power shortages.

The operational schedule includes a sequence of operational periods(steps), typically of a one hour length. During each step, it isnecessary to determine which generators should be on, and which shouldbe off, and how much electricity should be produced by each generatorthat is on.

The planning horizon, i.e., the duration of the schedule, is typicallybetween one day and one week. Finding the operational schedule that isoptimal among all possible schedules is a difficult computationalproblem due to the very large number of possible generator combinations,schedules that have to be considered, the differences in the operationalcosts of individual generators, the reliability and variations inoutput, and various existing operational constraints.

A large number of these constraints are temporal in nature, which turnsthe operational scheduling into a sequential decision making problem.For example, some generators have minimal and maximal on and off times,as well as limits on how fast the output of the generators can increaseor decrease. That is why turning a generator on or off has consequencesthat extend over long periods of time, and such decisions constitute acommitment to use (or not) the generator for multiple time steps.

For this reason, deciding which generators to turn on or off is commonlyknown as the unit commitment problem in power generation. After a set ofgenerators have been committed to be on at a specific moment in time,the optimal output to be produced by each generator has to bedetermined. Additional constraints must also be taken intoconsideration, such as the minimal and maximal output a generator canproduce. This nested optimization problem is known as an economicdispatch problem.

Given a set s of operational generators, which is a subset of allavailable generators, and a target electrical demand d, it is assumedthat F=ƒ(s, d) returns the total expected cost of producing theelectrical demand d by the generators in the set s, and G=g(s, d)returns an expected risk (probability) of not being able to meet thedemand with this set of generators. If the generators in the set scannot meet the demand d, for example because d exceeds the sum of theindividual maximal outputs of the generators in the set s, then it isassumed that the cost F is equal to the cost of running the generatorsat full capacity, and the risk G of failure to meet demand is one.

For most practical problems, the number of all possible schedules isprohibitively large to search exhaustively. If N generators areavailable, then there are 2^(N) possible subsets of on generators duringany time step. If there are a total of M time steps in the planninghorizon, e.g., M=24 for a planning horizon of one day and time step ofone hour, then the total number of all possible schedules is 2^(N,M).The tremendous combinatorial complexity of the operational schedulingproblem requires a more efficient computational method for anapproximate solution.

One simple method is to place all generators in a priority list orderedby the relative cost of output electricity per generator when operatingat maximal capacity, such that the generator with a lowest cost has ahighest priority. Given the expected demand d_(t) for the time step t,the available generators are operated according to the priority list,possibly committing new generators that were off if d_(t)>d_(t−1), orpossibly decommitting generators that were on if d_(t)<d_(t−1).

Minimal on and off times can be accommodated by modifying the prioritylist to exclude those generators that must be turned on or off tosatisfy these constraints. While feasible, such a method of operationalscheduling is far from optimal, and more advanced techniques based ondynamic programming, Lagrange relaxation, branch-and-bound, are known.

One approach decomposes the problem into stages corresponding to theindividual time steps of the schedule, and uses dynamic programming todetermine recursively the optimal cumulative cost-to-go, until the endof the schedule, for every feasible combination (subset) of generatorsfor the current stage.

Such a procedure reduces the computational complexity of the problem,because the computational complexity is linear in the number of stages(steps), and quadratic in the number of feasible combinations for everystage. However, the number of feasible combinations (2^(N)) is stillexponential in the number of available generators N. The heuristics toreduce that number of feasible combinations can possibly lead tosub-optimal solutions. Furthermore, if the state of a generator isrepresented by a Boolean variable (on/off), then it is not be possibleto accommodate requirements for minimum on and off times, and limits onramping rates.

When future power demand is completely known for the entire duration ofthe planning period, and the operator of the power generators has fullcontrol over how much electricity a generator generates after thegenerator has been turned on, the optimal operating schedule can bedetermined in advance, and executed accordingly as time advances.

In practice however, demand cannot be completely known. There are alwaysinaccuracies in forecasting, as well as random variations due to futureevents, e.g., higher load for air conditioners on a day that is warmerthan expected. Similarly, the output of generators cannot be completelyknown. For example, any generator can malfunction with some probability.In addition, the output of renewable power sources, such as photovoltaicpanels and wind turbines, can vary greatly, because the output isgoverned by uncontrollable natural forces.

Although less severe than complete generator malfunctions, thevariability of renewable power sources is an everyday reality, andaffects operational scheduling even more significantly. In the past, onepractical way to plan for deviations from expected demand and supply hasbeen to include a safety margin of extra capability to produce power bymeans of the committed generators, also called a spinning reserve. Thatis, the operational scheduling plans for a slightly higher power output.Determining how much this safety margin should be, and how it should bedistributed among the operational generators, is not an easy problem,and is subject to regulations.

Sometimes a rule of thumb is used to provide for a small safety marginof expected demand, e.g., 3%. In other cases, utilities have tocompensate for a possible loss of the largest generator. However, thatapproach is largely heuristic, and is not likely to work in the future,when renewable energy sources become more widespread. An alternativeapproach is to recognize that the uncertainty in power demand andgenerator supply make the problem stochastic, i.e., probabilistic andrandom, see e.g., U.S. Patent Application 20090292402, “Method &apparatus for orchestrating utility power supply & demand in real timeusing a continuous pricing signal sent via a network to home networks &smart appliances,” Nov. 26, 2009.

A stochastic operational scheduler determines a schedule that canaccommodate future variations of supply and demand, and provides asafety margin implicitly, by planning for all possible contingencies.One significant difficulty associated with that approach has been how torepresent all these possible contingencies, and how to plan for them.One model organizes all future possible realizations of the system(called scenarios) as a tree of scenario bundles. However, that modelfor representing stochasticity is limited to only a small number ofscenarios, whereas in a practical system the future can be realized inan infinite number of ways.

SUMMARY OF THE INVENTION

The embodiments of the invention provide a method for determining anoptimal conditional operational schedule for a set of power generatorsunder stochastic demand for electrical power, and stochastic output ofuncontrollable generators, e.g., renewable power sources such asphotovoltaic panels and wind turbines.

Unlike conventional operational schedules, which are fixed in advance, aconditional operational schedule depends on a future state of observablerandom variables (demand and output), and can result in different actualschedules depending on the observed outcomes for these variables. Thescheduler explicitly balances the operational cost of electricitygeneration with the risk of not being able to meet future electricitydemand.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow diagram of a method for determining an optimalconditional operational schedule for a set of N power generatorsaccording to embodiments of the invention; and

FIG. 2 shows precursor and successor states for controllable anduncontrollable power generators according to embodiment of theinvention.

FIG. 3 shows an AND/OR tree used to compute the optimal conditionaloperational schedule according to embodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

As shown in FIG. 1, the embodiments of the invention provide a method 50for determining an optimal conditional operational schedule 150 for aset of N power generators 100 under stochastic demand 101 for electricalpower, and stochastic output for a subset of the generators. The powergenerators supply power to consumers 105. The method can be implementedin a processor connected to a memory and input/output interfaces asknown in the art. The method uses a factored Markov decision process(fMDP) 130.

Input to the method includes the stochastic (random) demand d 101,generator related variables, such as operating costs and constraints102, and a risk coefficient α 103. The demand and generator relatedvariables are used to construct 110 the states and transitions of thefDMP. A cost function of the fMDP is also constructed 120 from thevariables 101-102 and the risk coefficient.

The method represents a power generation system including the multiplegenerators 100 by the fMDP 130. The fMDP represents a complex statespace using state variables and a transition model using a dynamicBayesian network (DBN) 131.

The method determines the optimal conditional operational schedule 150by solving 140 the fMDP using AND/OR trees 141, see FIG. 3.

Markov Decision Process

A Markov decision process (MDP) could be used to represent a system,wherein states evolve probabilistically over time. Typically, the MDP isdescribed by a four-tuple (X, A, R, P), where X is a finite set ofstates x, A is a finite set of actions a, R is a reward function suchthat R(x, a) represents the reward (i.e., cost) if action a is taken instate x; and P is a Markovian transition model, where P(x′|x, a)represents the probability of transitioning to a state x′ if action a istaken in state x.

The MDP could be used to represent the power generation system, forexample by representing each possible combination of the generators bythe state x, each combination of decisions regarding the state of thegenerators in the next time step by the action a, the cost of operatingthe generators in state x for the current period, and switching to asuccessor state x′ of the generators according to the action a at theend of the period according to the reward function R(x, a).

To make the transition function Markovian, the state of each generatoris not represented by a Boolean (on/off) variable, but by a multinomialvariable representing a number of time steps the generator has been onor off. This is necessary to ensure compliance with operatingconstraints regarding the minimal or maximal time a generator should beon or off.

If a generator should be on for at least (or at most, whichever isgreater) L time steps, and off for at least (or at most, whichever isgreater) l time steps, the state is represented by one of L+l values.Correspondingly, the state of the entire generation system that includesN generators can be one of (L+l)^(N) combinations. If planning is totake place over M time steps, then the total number of states |X| of theMDP is M(L+l)^(N).

However, for most practical problems, e.g., when L=l=5, N=20, M=24,|X|=24×10²⁰. Thus, the resulting conventional MDP is impossible tosolve, because existing exact methods for solving the MDP arecomputationally feasible only when |X| is limited to several millionstates. In addition, the MDP is cumbersome to construct and maintain.

Factored Markov Decision Process

Therefore, the method according to the preferred embodiments of theinvention uses the fMDP 130. In the fMDP, the set of states of theprocess is implicitly described by an assignment to a set of individualrandom variables X={X₁, X₂, . . . , X_(n)}, where each state variableX_(i) has values in a finite domain Dom(X_(i)). That is, an individualstate x is also a set of assignments {x₁, x₂, . . . , x_(n)}, such thatx_(i)ε Dom(X_(i)).

Dynamic Bayesian Network

The transition model can be represented compactly by means of the DBN131. The DBN represents an evolution of a probabilistic system, i.e.,the power generators, from a one time step t to a next time step t+1. IfX={X₁, X₂, . . . , X_(n)} is the precursor state of the system at thefirst time step, and X′={X′₁, X′₂, . . . , X′_(n).} is the successorstate of the system at the next time step, then the DBN τ has 2n randomvariables in the set {X₁, X₂, . . . , X_(n), X′₁, X′₂, . . . , X′_(n)},typically organized in two layers, i.e., a precursor layer and asuccessor layer.

The transition graph of the DBN τ can be represented by a two-layerdirected acyclic graph, wherein the nodes are the 2n random variables.The parents of node X′_(i) in the graph of the BDN τ are denoted byParents_(τ)(X′_(i)). Furthermore, a conditional probability distribution(CPD) is defined for variable X′_(i), such that the CPD is conditionalonly on the variables inParents_(τ)(X′_(i)):P_(τ)[X′_(i)|Parents_(τ)(X′_(i))]. Then, the entiretransition function for the fMDP can be factored as the product of theCPD of individual variables X′_(i):P_(τ)(x′|x)=Π_(i)P_(τ)(x′_(i)|u_(i)),where u_(i) is an assignment of Parents_(τ)(X′_(i)) in the value of thestate x.

To handle multiple actions a, either a separate DBN is constructed foreach action, or individual action variables are included in the DBN,such that the CPD of the variables depend on the action variables, or asubset of the variables.

fMDP Variables

For the purposes of representing a power generation system 100 by thefMDP 130, the following variables are used. The number n of individualstate variables is equal to the number of generators N plus one, i.e.,n=N+1. Each individual variable X_(i) represents the state of onegenerator, and the last variable represents the stochastic power demand,which is also a random variable.

Generator Types

As shown in FIG. 2, there are many types of generators: controllable,e.g., coal, gas, oil, hydroelectric, nuclear, and uncontrollable, e.g.,solar and wind. These two types of generators are representeddifferently in the DBN of the system. FIG. 2 also shows steps at time t201 and t+1 202, discretized demand 210, the states 220 of theuncontrolled generators, the states 230 of the controllable generators,statistical dependences 240, and a decision (action a) to turn agenerator on or off.

Controllable Generators

For a controllable generator i that satisfies constraints regardingminimal/maximal on and off times, as described above, the variable X_(i)can take on L+l possible different values in the domain setDom(X _(i))={(on, 1), (on, 2), . . . , (on, L), (off, 1), (off, 2), . .. , (off, l)}.

In the DBN 131, there are two parent nodes of X′_(i). One parent node isthe state X_(i) of the generator in the precursor state at time t, andthe other parent node is the Boolean variable a_(i) that represents theaction to turn generator i on or off in the successor state at time t+1.

The CPD P_(τ)(X′_(i)|a_(i)) of the variable X′_(i) represents the timeevolution of the state of the generator, e.g., state (on, 1) is followedby (on, 2), if a_(i)=on, or by (off, 1), if a_(i)=off, with probabilityone, subject to the operating constraints, e.g., (on, L) is followed by(off, 1), regardless of the action a_(i), when the generator can stay onfor at most L time periods. Furthermore, a probability of malfunctioncan be added to the CPD, such that the state of the generator becomes(off, 1) with a probability equal to the likelihood of malfunctionwithin one time period, regardless of the action a_(i).

Non-Controllable Generators

A non-controllable generator is always on, but has stochastic variationin output power due to, e.g., varying weather conditions such as windand sunlight. The random variable X_(i) represents a difference ΔE=E−Ēbetween the actual power output E of the generator and the forecast Ēfor its output that is available in advance at the time of scheduling.Note that E and ΔE are random variables with subsequent observed valuesthat are not known at the time of scheduling, whereas Ē is a knownconstant at that time. Normally, the difference ΔE is a continuousvariable, and a suitable discretization is performed, either by binningthe difference into several discrete intervals, or using a more advanceddiscretization scheme.

In the DBN, the variable X′_(i) for such a non-controllable generatorhas only one parent node X_(i), and the corresponding CPDPτ(X′_(i)|x_(i)) can be constructed in several possible ways. Onepossible way is to observe experimental data, and assign theprobabilities in the CPD such that Pτ(X′_(i)|x_(i))=F(Δe′|Δe), whereF(Δc′|Δc) is the frequency of observing difference Δc′ when thedifference in the previous time step was Δe.

Another way is to assume that the output of the generator is adiscrete-time auto-regressive stochastic process of order 1, i.e.,(AR(1)), and estimate a single regression coefficient ρ fromexperimentally observed T residuals Δe_(t), such that Δe_(t+1)=ρΔe_(t),for t=1, . . . , T. A suitable value for ρ can be obtained by linearregression so that the equality is satisfied in a least-square sense.After the regression coefficient ρ is obtained, P_(τ)(X′_(i)|x_(i)) canbe determined by means of discretizing the AR(1) process. The lastvariable, X_(N+1), which represents the probabilistic evolution of powerdemand, is treated analogously to those representing non-controllablegenerators. Alternatively, a continuous-time mean-reversal stochasticprocess can be used.

Reward Function

The reward function R(x, a) of the fMDP is determined as follows. Givena value of the state variable x, let s be the subset of all generators,which are on during the state x. Furthermore, let d be the amount ofpower demand that corresponds to the demand variable x_(N+1) within x.Then, after solving the economic dispatch problem for this subset ofgenerators and target demand, let F=ƒ(s, d) be the cost of meeting thedemand with the current set of generators. Furthermore, let G=g(s, d) bethe risk, that is, the probability of not being able to meet demand dwith the generators in the subset s.

For specific known values of s and d, this risk is completely known,that is, either zero or one. The cost of changing the state of thegenerators according to the action a at the end of the current time stepis H. Then, for a particular risk coefficient α 103, the total reward orcost isR(x,a)=F+H+αG.

Solving the fMDP

By specifying all elements of the fMDP, the problem of determining theconditional operational schedule 150 for the power generators 100 isreduced to that of solving the fMDP. The fMDP can be solved by any ofknown methods for approximate dynamic programming and approximate linearprogramming The solution is a policy that maps every state of the fMDPto an action within the available set of actions, such that theexecution of this action maximizes the defined reward, i.e., minimizesthe cost. This policy is the conditional operational schedule for thepower generators, and if followed, the policy determines which generatorto turn on and off at every time step t, depending on the states of thepower generators at the beginning of that time step. Thus, the cost ofmeeting demand and the risk of not being able to meet the demand arejointly minimized over the planning horizon of M time steps, accordingto the risk coefficient α 103, which can be user defined.

As shown in FIG. 3, one specific approximate method for solving the fMDPis to restrict the number of states in the fMDP to a reasonable subset,and use AND/OR trees 141 to find the optimal conditional schedule 150.The AND/OR tree include two types of nodes: AND nodes 301, 303, 305, andOR nodes 302 and 304.

The AND nodes represent states that the system can be in at thebeginning of a decision period. In this case, the system is described bythe triple (u_(t), x_(t), d_(t)), where u_(t) is the configuration (on,off) for all generators at the beginning of time period t, x_(t) is thestate of the controllable generators of the MDP at that time, and d_(t)is the net demand observed at that time, computed as the differencebetween the total demand and the output of the uncontrollable powersources. The OR nodes represent decisions that can be made. In thiscase, the decisions are the configurations u_(t) that can be chosen attime t.

The root node 301 of the AND/OR tree is always an AND node, andrepresents the initial state of the system at the time of computing theschedule (t=0). The OR nodes at the second level 302 are the possibleconfigurations that can be chosen for the beginning of the first timeperiod (t=1). The net demand d at the beginning of that period is arandom variable, and can take on several values in the set {d₁, d₂, . .. } with various probabilities, as described by the transitionprobabilities in the DBN 131. Which one demand is taken will becomeclear at the beginning of the first decision period. This is representedas having multiple descendant AND nodes 303 of the OR node 302 in theAND/OR tree. The tree is expanded further down by adding descendant ORnodes for each possible AND node, etc., until depth equal to theplanning horizon is reached.

The optimal schedule can then be computed by means of dynamicprogramming, as follows. Let V(u_(t+1)|u_(t), x_(t), d_(t)) be the valueof an OR node corresponding to the configuration decision intended fortime t+1, taken when the system is in state (u_(t), x_(t), d_(t)) inperiod t. Also, let V(u_(t), x_(t), d_(t)) be the value of the AND nodecorresponding to that state. Then, the following two dynamic programmingequations can be applied in a bottom-up manner, starting from the leavesof the tree and proceeding up to the root of the tree, to determine thevalue functions of all nodes in the tree:

$\mspace{20mu}{{V\left( {u_{t},x_{t},d_{t}} \right)} = \left\{ {{{\begin{matrix}{{R\left( {x_{T},u_{T}} \right)},{{{when}\mspace{14mu} t} = T}} \\{{\min\limits_{u_{t + 1}}{V\left( {\left. u_{t + 1} \middle| u_{t} \right.,x_{t},d_{t}} \right)}},{otherwise}}\end{matrix}{V\left( {\left. u_{t + 1} \middle| u_{t} \right.,x_{t},d_{t}} \right)}} = {{R\left( {x_{t},u_{t}} \right)} + {\sum\limits_{d_{t + 1}}{{\Pr\left( d_{t + 1} \middle| d_{t} \right)}{V\left( {u_{t + 1},x_{t + 1},d_{t + 1}} \right)}}}}},} \right.}$Where T is the terminal (or last) decision step, and x_(t+1) is thestate of the controllable generators that would be assumed ifconfiguration u_(t+1) is chosen for time period t+1.

Once the values of all nodes have been computed, the optimal schedulecan be executed as follows. Starting from the initial state 301 (u₀, x₀,d₀), the system is operated in configuration u₀. For the next decisionperiod, the scheduler chooses the configuration u₁ that corresponds tothe OR node 302 with the lowest value function:

$u_{1} = {\underset{u_{1}}{\arg\;\min}{V\left( {\left. u_{1} \middle| u_{0} \right.,x_{0},d_{0}} \right)}}$

Then, depending on the observed net demand d₁ for the first period, thesystem transitions to one of the AND nodes 303. The choice ofconfiguration then proceeds analogously, always choosing theconfiguration for the next time period according to:

$u_{t + 1} = {\underset{u_{t + 1}}{\arg\;\min}{V\left( {\left. u_{t + 1} \middle| u_{t} \right.,x_{t},d_{t}} \right)}}$until the end of the planning horizon is reached.

Although this computational method always finds the optimal conditionalschedule, it is very complex computationally, because the number ofpossible configurations that can be chosen at any time period is on theorder of 2^(N). Furthermore, the tree also branches on the possiblevalues of net demand. In practice, the branching factor of the tree mustbe limited to a reasonable number, in order to make the methodcomputationally feasible.

One possible method for limiting the branching factor of the tree is toconsider only a small subset of all possible configurations of thegenerators as candidates for each step. As noted above, a priority listof generators can be used, such that the number of candidateconfigurations is only N+1.

Another method is to use a known process for generating deterministicschedules, and compute optimal schedules for target demand that variesfrom expected demand by a given percentage, for example from −10% to+10%. The deterministic scheduler then finds sequences of suitableconfigurations for each time period and each level of demand. By placingthe configurations for the same time period into a single candidate setU_(t) for the configurations u_(t), the branching factor of the AND/ORcan be limited significantly, and restricted only to configurations thatare suitable for likely variations of electricity demand.

Although the invention has been described by way of examples ofpreferred embodiments, it is to be understood that various otheradaptations and modifications may be made within the spirit and scope ofthe invention. Therefore, it is the object of the appended claims tocover all such variations and modifications as come within the truespirit and scope of the invention.

We claim:
 1. A method for determining an optimal conditional operationalschedule for a set of power generators, comprising the steps of:constructing states and transitions of a factored Markov decisionprocess (fMDP) from a target electrical demand and generator variables,wherein the states of the fMDP indicate a number of time steps a stateof each power generator in the set is ON or OFF, wherein each transitionof the fMDP changes the state of a power generator between ON and OFFstates, and wherein the set of power generators includes multiple powergenerators; constructing a cost function for the fMDP based on theelectrical demand, the generator variables, and a risk coefficient, suchthat the cost function is a weighted sum of a cost of meeting theelectrical demand and a risk of not meeting the electrical demand; andsolving the fMDP to obtain the optimal conditional operational schedule,wherein the steps are performed in a processor.
 2. The method of claim1, wherein the demand is stochastic.
 3. The method of claim 1, whereinthe set of generators includes uncontrollable generators, wherein eachuncontrollable generator has a stochastic output.
 4. The method of claim1, wherein the generator variables include a number of the generators,costs, and constraints for operating the generators.
 5. The method ofclaim 1, wherein the fMDP is represented by a dynamic Bayesian network(DBN).
 6. The method of claim 5, wherein the DBN represents an evolutionof the power generators from a time step t to a next time step t+1, andwherein X={X₁, X₂, . . . , X_(n)} is a precursor state at the time t,and X′={X′₁, X′₂, . . . , X′_(n)} is a successor state at the time t+1.7. The method of claim 5, wherein the DBN is represented by a two-layerdirected acyclic graph.
 8. The method of claim 1, wherein the solvinguses approximate dynamic programming.
 9. The method of claim 1, whereinthe states of the fMDP are represented by a set of random variablesX={X₁, X₂, . . . X_(n)}, where each state variable X_(i) has values in afinite domain Dom(X_(i)), and an individual state x has a set ofassignments {x₂, x_(n), . . . , x_(n)}, such that x_(i) εDom(X_(i)). 10.The method of claim 1, wherein the solving of the fMDP uses an AND/ORtree and dynamic programming.
 11. The method of claim 10, whereinbranching of the AND/OR tree is limited to a subset of suitableconfigurations of the controllable generators.
 12. The method of claim11, wherein the subset of suitable configurations of the controllablegenerators is constructed by priority list for the generators.
 13. Themethod of claim 11, wherein the subset of suitable configurations of thecontrollable generators is constructed by executing deterministicschedulers for varying levels of demand around an expected value of thedemand, and observing which configurations are used in the optimalconditional operational schedules.
 14. The method of claim 11, wherein anet demand is computed by subtracting the output of all uncontrollablevariables from a total demand.
 15. The method of claim 11, wherein thedemand variable is discretized and limited to only a number of possiblediscrete values.
 16. The method of claim 15, wherein a probabilistictransition function of the net demand variable is estimated from adiscrete-time auto-regressive stochastic process.
 17. The method ofclaim 15, wherein the probabilistic transition function of the netdemand variable is estimated from a continuous-time mean-reversalstochastic process.
 18. The method of claim 1, wherein the cost functionR(x, a) isR(x,a)=F+H+αG, wherein F is the cost of meeting the electrical demand, His a cost of changing the state x of the power generator according to anaction a, G is the risk of not meeting the electrical demand, and α isrisk coefficient.